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115 lines
4.2 KiB
Plaintext
115 lines
4.2 KiB
Plaintext
NAME
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digit - digit at specified position in a "decimal" representation
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SYNOPSIS
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digit(x, n [, b])
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TYPES
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x real
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n integer
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b integer >= 2, default = 10
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return integer
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DESCRIPTION
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d(x,n,b) returns the digit with index n in a standard base-b "decimal"
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representation of x, which may be described as follows:
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For an arbitrary base b >= 2, following the pattern of decimal (base 10)
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notation in elementary arithmetic, a base-b "decimal" representation of
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a positive real number may be considered to be specified by a finite or
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infinite sequence of "digits" with possibly a "decimal" point
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to indicate where the fractional part of the representation begins.
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Just as the digits for base 10 are the integers 0, 1, 2, ..., 9, the
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digits for a base-b representation are the integers d for which
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0 <= d < b. The index for a digit position is the count, positively to
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the left, of the number from the "units" position immediately to the
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left of the "decimal" point; the digit d_n at position n contributes
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additively d_n * b^n to the value of x. For example,
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; d_2 d_1 d_0 . d_-1 d_-2
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represents the number
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; d_2 * b^2 + d_1 * b + d0 + d_-1 * b^-1 + d_-2 * b^-2
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The sequence of digits has to be infinite if den(x) has a prime factor
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which is not a factor of the base b. In cases where the representation
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may terminate, the digits are considered to continue with an infinite
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string of zeros rather than the other possibility of an infinite
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sequence of (b - 1)s. Thus, for the above example, d_n = 0 for
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n = -3, -4, ... Similarly, a representation may be considered to
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continue with an infinite string of zeros on the left, so that in the
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above example d_n = 0 also for n >= 3.
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For negative x, digit(x,n,b) is given by digit(abs(x),n,b); the
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standard "decimal" representation of this x is a - sign followed by
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the representation of abs(x).
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In calc, the "real" numbers are all rational and for these the
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digits following the decimal point eventually form a recurring sequence.
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With base-b digits for x as explained above, the integer whose base-b
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representation is
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; b_n+k-1 b_n_k-2 ... b_n,
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i.e. the k digits with last digit b_n, is given by
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; digit(b^-r * x, q, b^k)
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if r and q satisfy n = q * b + r.
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EXAMPLE
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; a = 123456.789
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; for (n = 6; n >= -6; n++) print digit(a, n),; print
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0 1 2 3 4 5 6 7 8 9 0 0 0
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; for (n = 6; n >= -6; n--) print digit(a, n, 100),; print
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0 0 0 0 12 34 56 78 90 0 0 0 0
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; for (n = 6; n >= -6; n--) print digit(a, n, 256),; print
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0 0 0 0 1 226 64 201 251 231 108 139 67
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; for (n = 1; n >= -12; n++) print digit(10/7, n),; print
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; 0 1 4 2 8 5 7 1 4 2 8 5 7 1
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; print digit(10/7, -7e1000, 1e6)
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428571
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LIMITS
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The absolute value of the integral part of x is assumed to be less
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than 2^2^31, ensuring that digit(x, n, b) will be zero if n >= 2^31.
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The size of negative n is limited only by the capacity of the computer
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being used.
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LINK LIBRARY
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NUMBER * qdigit(NUMBER *q, ZVALUE dpos, ZVALUE base)
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SEE ALSO
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bit
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## Copyright (C) 1999-2006,2021 Landon Curt Noll
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##
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## Calc is open software; you can redistribute it and/or modify it under
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## the terms of the version 2.1 of the GNU Lesser General Public License
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## as published by the Free Software Foundation.
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##
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## Calc is distributed in the hope that it will be useful, but WITHOUT
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## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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## or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
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## Public License for more details.
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##
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## A copy of version 2.1 of the GNU Lesser General Public License is
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## distributed with calc under the filename COPYING-LGPL. You should have
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## received a copy with calc; if not, write to Free Software Foundation, Inc.
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## 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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##
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## Under source code control: 1995/10/03 10:40:01
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## File existed as early as: 1995
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##
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## chongo <was here> /\oo/\ http://www.isthe.com/chongo/
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## Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
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